Minimal charts of type (3,3) are equivalent to a specific subchart.
problem Characterizing minimal charts of type (3,3).
method Analyzing subgraphs and C-move equivalence.
result Minimal charts of type (3,3) are equivalent to a specific subchart.
In this paper, we shall show a condition for that a chart is C-move equivalent to the product of two charts, the union of two charts Γ∗ and Γ∗∗ which are contained in disks D∗ and D∗∗ with D∗∩D∗∗=∅.
A formula for the difference of Vassiliev invariants of degree k+1 of two knots all of whose Vassiliev invariants of degree k agree is proven. The proof uses K. Habiro's C-moves and his theorem which relates them to Vassiliev invariants.
Study examines equivalence relations on the pair of pants, proving k-equivalence implies 1-equivalence and 2-equivalence.
problem Understanding equivalence classes of closed curves on the pair of pants.
method Examined k-equivalence, proving it implies 1-equivalence and 2-equivalence, and deepened understanding of 1-equivalence.
result k-equivalence implies 1-equivalence and 2-equivalence on the pair of pants.
Study geometric equivalence of smooth map germs.
problem Equivalence relations among smooth map germs with respect to G-structures.
method Generalization of right-left equivalence (A-equivalence) to include geometric structures.
result Interesting applications of these equivalence relations.
Paper solves equivalence problems for fifth-order differential operators using Cartan's method.
problem Equivalence problem for fifth-order differential operators under fiber-preserving transformations.
method Cartan's method of equivalence applied to solve two versions of the equivalence problem.
result Sufficient and necessary conditions for fiber-preserving transformations between fifth-order differential operators.
Paper extends Stanford's equivalence to virtual knots and proves equivalence to Goussarov-Polyak-Viro's n-equivalence.
problem Characterizing finite type invariants in virtual knot theory.
method Using the lower central series of the pure virtual braid group to define an L_n-equivalence.
result L_n-equivalence on virtual string links is equivalent to Goussarov-Polyak-Viro's n-equivalence.
Geodesically equivalent metrics on homogeneous spaces are shown to be affinely equivalent.
problem Characterizing geodesically equivalent metrics on homogeneous spaces.
method Analyzing left G-invariant metrics on G/H and using algorithms to find geodesically equivalent metrics. result Existence of non-proportional geodesically equivalent metrics implies non-full holonomy algebra.
In this paper we discuss four problems regarding Markov equivalences for subclasses of loopless mixed graphs. We classify these four problems as finding conditions for internal Markov equivalence, which is Markov equivalence within a subclass, for external Markov equivalence, which is Markov equivalence between subclas…
This paper shows that certain knots are equivalent under a specific relation.
problem Understanding the equivalence classes of genus one knots.
method Using knot Floer complexes and Heegaard Floer theory invariants.
result Any genus one knot is ν+-equivalent to a trefoil, its mirror, or the unknot. This article is dedicated to solve the equivalence problem for two third order differential operators on the line under general fiber--preserving transformation using the Cartan method of equivalence. We will do three versions of the equivalence problems: first via the direct equivalence problem, second equivalence pro…
Polynomials with similar modulus levels are bilipschitz equivalent.
problem Understanding when polynomials are bilipschitz equivalent.
method Examined polynomials with moduli up to bilipschitz equivalence.
result Level curves of polynomials are bilipschitz equivalent.
Equivalent bicategories constructed from action Lie groupoids.
problem Equivalence of bicategories constructed from action Lie groupoids.
method Localizing at equivariant weak equivalences, surjective submersive equivariant weak equivalences, and all weak equivalences.
result Weak equivalences between action Lie groupoids are isomorphic to compositions of nice forms of equivariant weak equivalences.
Paper discusses when virtual links are equivalent as twisted links.
problem Determining equivalence of virtual links as twisted links.
method Using stable equivalence classes of links in oriented thickenings of surfaces.
result Necessary and sufficient condition for virtual links to be equivalent as twisted links.
Develops L∞ spaces over dg manifolds and establishes an equivalence with L∞ algebroids.
problem Defining and comparing L∞ spaces and algebroids over dg manifolds. method Establishes an equivalence between categories of L∞ algebroids and L∞ spaces, constructs a faithful functor. result Detects weak equivalences between L∞ algebroids and L∞ spaces. Two Riemannian manifolds are called eigenvalue equivalent when their sets of eigenvalues of the Laplace-Beltrami operator are equal (ignoring multiplicities). They are (primitive) length equivalent when the sets of lengths of their (primitive) closed geodesics are equal. We give a general construction of eigenvalue equ…
Develops a new method for solving equivalence problems in pseudo-groups.
problem Solving equivalence problems in pseudo-groups.
method Combining Cartan's equivalence method and equivariant moving frame for pseudo-groups.
result A hybrid equivalence method that extends and illuminates its two progenitors.
Unified notion of equivalence for causal graphs, cyclic or acyclic.
problem Defining equivalence for causal graphs with cycles.
method General unified notion of equivalence based on distribution generation.
result Unified notion of equivalence and quasi-equivalence for causal graphs.
Paper refines braidoid equivalence for spherical knotoids.
problem Equivalence of knotoid diagrams in spherical space.
method Refined L-equivalence of braidoid diagrams. result Equivalence theorem for multi-knotoid diagrams in S2. In this paper, two sufficient conditions are provided for given two K-equivalent map-germs to be bi-Lipschitz A-equivalent. These are Lipschitz analogues of the known results on C^r-A-equivalence (0≤r≤∞) for given two K-equivalent map-germs. As a corollary of one of our results, a Lipschitz version of …
New Upsilon invariants rule out stable equivalence of knot complexes.
problem Stable equivalence of knot complexes and its invariants.
method Secondary Upsilon invariants defined by Kim and Livingston.
result Relations between Upsilon invariants do not extend to stable equivalence.
Paper explores state-action equivalence in RL, improving regret bounds.
problem Improving reinforcement learning performance by leveraging state-action equivalence.
method Introduces a notion of similarity between state-action pairs, defines equivalence structure, and presents algorithms for confidence sets.
result Confidence sets improve RL performance, especially in known equivalence structures.
New R-equivalence classes found for torus knot diagrams.
problem Classifying colorings of torus knots.
method Introducing R-equivalence relation on quandle colorings.
result Determined R-equivalence classes for RotE2-colorings of torus knots. In present paper, the equivalence problem for fourth order differential operators with one variable under general fiber-preserving transformation using the Cartan method of equivalence is applied. Two versions of equivalence problems are considered. First, the direct equivalence problem and second equivalence problem i…
Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Delta-equivalent) to a trivial link. We study link-homotopy and self Delta-equivalence on a certain componen…
Paper defines curvature equivalence for Legendre curves in a plane.
problem No specific problem stated; focuses on Legendre curves.
method Introduced curvature equivalence relation for Legendre curves.
result Local and global classifications of Legendre curves under curvature equivalence.
The equivalence (or weak equivalence) classes of orientation-preserving free actions of a finite group G on an orientable 3-dimensional handlebody of genus g can be enumerated in terms of sets of generators of G. They correspond to the equivalence classes of generating n-vectors of elements of G, where n=1+(g-1)/|G|, u…
Let M and N be Nash manifolds, and f and g Nash maps from M to N. If M and N are compact and if f and g are analytically R-L equivalent, then they are Nash R-L equivalent. In the local case, Cinfty R-L equivalence of two Nash map germs implies Nash R-L equivalence. This shows a difference of Nash…
Study stable equivalence relations on 4-manifolds, proving homotopy equivalent manifolds with abelian fundamental group are stably diffeomorphic.
problem Classifying stable equivalence relations on 4-manifolds.
method Combination of modified and classical surgery, focusing on homotopy equivalence up to stabilisation.
result Closed oriented homotopy equivalent 4-manifolds with abelian fundamental group are stably diffeomorphic.
Study shows equivalence in foliations and pre-symplectic forms aligns with gauge equivalence.
problem Deformation theory of foliations and pre-symplectic forms.
method Proved geometric equivalence agrees with algebraic gauge equivalence using L∞-algebras. result Gauge equivalences for foliations and pre-symplectic structures are consistent.
The paper studies Lipschitz equivalence of Cantor sets via polynomial irreducibility.
problem Lipschitz equivalence of Cantor sets.
method Analyzing the irreducibility of polynomials.
result Two Cantor sets are Lipschitz equivalent if their contraction vectors are equivalent.
Establishes equivalence between models of derived stacks.
problem Tackles the equivalence between different models of derived geometry.
method Uses Quillen equivalence to show categories of higher derived stacks are equivalent.
result Shows equivalence among models of derived manifolds, Carchedi-Roytenberg, Behrend-Liao-Xu, and Alexandrov-Kontsevich-Schwarz-Zaboronsky.
Handlebody groups are rigid under measure equivalence.
problem Proving handlebody groups are rigid under measure equivalence.
method Proving superrigidity for measure equivalence of handlebody groups.
result Every countable group measure equivalent to handlebody groups is virtually isomorphic to them.
Complex equivalence classes found in graph homotopy.
problem Complexity of proper homotopy equivalence in graphs.
method Demonstrated Borel completeness and comeager equivalence classes.
result Complex equivalence classes exist in infinite graphs.
If the fundamental group of the complement of a smooth embedding f: S^2 \subset R^4 is a cyclic group, the map can be deformed to the standard embedding by a generic one-parameter family with at most cusp singularities. If two smooth embeddings are connected by such a deformation, they will be called cusp equivalent. W…
Characterizes Bayesian networks up to unconditional equivalence.
problem Characterizing Bayesian networks up to unconditional equivalence.
method Transformational characterization via undirected graphs and specified moves.
result Two DAGs are in the same UEC if and only if one can be transformed into the other via a finite sequence of moves.
We study gauge transformations of Dirac structures and the relationship between gauge and Morita equivalences of Poisson manifolds. We describe how the symplectic structure of a symplectic groupoid is affected by a gauge transformation of the Poisson structure on its identity section, and prove that gauge-equivalent in…
System uses neural networks to prove program equivalence via rewrite rules.
problem Proving equivalence between two dataflow graphs.
method Developed a graph-to-sequence neural network trained on example generation to find semantics-preserving rewrite rules.
result System correctly outputs a rewrite sequence for 96% of program pairs, proving equivalence.
Homotopy equivalence of cotangent bundles' function algebras is shown.
problem Understanding homotopy equivalence in cotangent bundles and their function algebras.
method Using shifted Poisson algebras and homotopy equivalence of bundles.
result Homotopy equivalent bundles have equivalent Poisson algebras.
New equivalence for singular foliations preserves transverse geometry.
problem Transverse geometry of singular foliations.
method Introducing a new notion of equivalence for singular foliations and showing compatibility with holonomy groupoids.
result Unified invariants and equivalence of singular foliations.
Equivalence proven between different Ricci curvature definitions.
problem Equivalence of various Ricci curvature definitions.
method Proof of equivalence between different Ricci curvature definitions.
result Equivalence proven between several Ricci curvature definitions.
Smooth orbit equivalence proves metric equivalence for geodesic flows.
problem Proving metric equivalence for geodesic flows under orbit equivalence.
method Proving metric equivalence for geodesic flows under orbit equivalence.
result Smooth orbit equivalence implies conformal equivalence of metrics.
Dynamic systems linked to infinite permutation matrices.
problem Dynamic equivalence of control systems.
method Association of infinite permutation matrices.
result Relationship between dynamic equivalences and permutation matrices.
This paper classifies superintegrable systems on 2D geometries with projective symmetries.
problem Classifying superintegrable systems on 2D geometries with projective symmetries.
method Combining metric projective differential geometry and superintegrability, defining projective equivalence, and applying transformation rules.
result Potentials of projectively equivalent Hamiltonians follow a linear superimposition rule.
Linearizability of singular foliations is preserved under a specific equivalence relation.
problem Preserving properties of singular foliations under equivalence relations.
method Characterization of tubular neighborhood embeddings using Euler-like vector fields.
result Linearizability along a leaf is a Morita invariant.
Establishes Morita equivalence for Nijenhuis structures and proves invariance of modular class.
problem Morita equivalence for Nijenhuis structures and Poisson-Nijenhuis manifolds.
method Global-to-infinitesimal correspondence using Lie functor and enhanced known equivalences.
result Modular class of Poisson-Nijenhuis manifolds is invariant under Morita equivalence.
Equivalence proven for isocapacitary mass notions.
problem Proving equivalence of isocapacitary mass notions.
method Proof of equivalence for G. Huisken's and J. L. Jauregui's isocapacitary mass.
result Equivalence of isocapacitary mass notions proven.
The paper compares isotopic and diffeomorphic links in lens spaces.
problem Comparing isotopic and diffeomorphic links in lens spaces.
method Provides a set of moves on disk, band, and grid diagrams to connect diffeomorphic links.
result There are up to four isotopy equivalent links in each diffeo equivalence class.